What is the purpose of an integral in calculus?

[delve]

Could someone tell me what an integral actually does? I understand that it has something to do with adding infinitely many things, but as you can see, this understanding is vague at best. Please help me. Thanks.

 

[Char. Limit]

Integration is a method of calculating infinitely many slivers of infinitely tiny areas between a function and the x-axis, then summing up these slivers. Indefinite integration has been proven to be the opposite of differentiation.

 

[delve]

Ok, this makes sense, thank you! :)

 

[paulfr]

Integration is just multiplication when one of the inputs is changing.

Remember that Calculus is the Mathematics of Change.

Integral of 3 dx over the interval from 0 to 4 is = 3 x [ 4 - 0 ] = 12

Now suppose we want to do this multiplication for y = f (x) = 2x + 5 rather than y = f(x) = 3
We have to divide up the area
[recall the 2 dimensional interpretation of integration as the area under a curve]
in to small slices and add up all the areas.
This is consistent with the fact that multiplication is nothing more than repeated addition.
3 x 4 = 12
3 x 4 = 3 + 3 + 3 + 3
3 x 4 = 4 + 4 + 4

The smaller/narrower the slices [ dx ] the more accurate the result.
In the limit as delta x goes to zero we have the integral which is the product of f(x) and dx over the interval.

Hope that helps